Computational fluid dynamics applied to membranes: State of the art and opportunities

Chemical Engineering and Processing 45 (2006) 437–454

Computational fluid dynamics applied to membranes: State of the art and opportunities R. Ghidossi a , D. Veyret a , P. Moulin b,∗ a

Institut Universitaire des Syst`emes Thermiques Industriels (IUSTI CNRS-UMR 6595), Technopˆole de Chˆateau Gombert, 5 rue Enrico Fermi, 13453 Marseille Cedex 13, France b Laboratoire d’Etudes et d’Applications des Proc´ ed´es S´eparatifs (LPPE CNRS-UMR 6181), Universit´e Paul C´ezanne d’Aix Marseille, Europˆole de l’Arbois, Bˆatiment Laennec, Hall C, BP 80, 13545 Aix en Provence Cedex 04, France Received 5 July 2005; received in revised form 3 November 2005; accepted 3 November 2005 Available online 13 December 2005

Abstract Membrane filtration has become firmly established as a primary technology for ensuring the purity, safety and/or efficiency of the treatment of water or effluents. In this paper, we review the improvements that have been achieved concerning the membranes used for microfiltration, ultrafiltration, nanofiltration/reverse osmosis processes during the last decades. More especially, we review the state of the art computational fluid dynamics (CFD) methods applied to membranes processes. Many studies have focused on the best ways of using a particular membrane process. But, the design of new membrane systems requires a considerable amount of process development as well as robust methods. Computational fluid dynamics may provide a lot of interesting information for the development of membrane processes. We review the different ways in which CFD methods are used to improve membrane performance. © 2005 Elsevier B.V. All rights reserved. Keywords: Computational fluid dynamics; Membrane; Review; Opportunities; Flow

1. Introduction Over the past two decades, membrane filtration processes have played a more and more important role in industrial separation. A number of studies have focused on enhancing microfiltration, ultrafiltration, nanofiltration and reverse osmosis processes. These include: (i) testing of new membrane materials; (ii) use of different pore sizes; (iii) determination of conditions for optimal selectivity; (iv) attempts to determine the optimum trans-membrane pressure or permeate flux to minimize fouling. The design of a new membrane typically requires a considerable amount of process development. The definition of a generic model with applicability to membrane systems requires rigorous and robust methods. Numerous improvements of the technology have allowed membrane selection for a particular process to be done more easily and more quickly. Typically, the development of the membrane is done

∗

Corresponding author. Tel.: +33 4 42 90 85 05; fax: +33 4 42 90 85 15. E-mail address: [email protected] (P. Moulin).

0255-2701/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2005.11.002

sequentially, with the use of a statistical design methodology to minimize the number of experiments needed to explore a wide range of variables. However, there are several aspects of this constantly evolving technology that have not yet been resolved and still pose an obstacle toward its development. One of these important aspects is the understanding of membrane fouling and subsequent permeate flux decline, which is inevitably associated with these processes. In this review, we describe the techniques that are available for the optimization of these processes. The modelling of flow and concentration polarisation in membrane systems or thin channels with permeable walls is not new and computational fluid dynamics (CFD) is an important tool used to develop membrane processes. We summarise the methods used to enhance microfiltration, ultrafiltration, nanofiltration and reverse osmosis processes. In the first part, we describe the CFD results for laminar and turbulent conditions and more especially for the different models used to describe mass transfer. In the second part, we study the CFD approaches used to understand the decrease in membrane fouling by using different instabilities or turbulence promoters. We try to describe the opportunities that can be used for the optimization of membrane

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processes. For the sake of clarity, this review will not detail all the assumptions made and the equations used in each study. 2. CFD and membrane models 2.1. Computational fluids dynamics The development of this technology is due to the increasing number of different types of applications of these processes in different domains, particularly in the industrial sector. Membrane filtration can be used in a broad range of applications [1]. A membrane is a permselective barrier of a few hundred nanometers to a few millimetres in thickness, which under the effect of a transfer force, will allow or prohibit the passage of certain components between two separate mediums. There are two main models making it possible to better comprehend this phenomenon. The first model, developed by Kadem and Katchalsky, is a solubilization-diffusion model based on the concept of capillarity. The second one is the model of polarization that results in a progressive accumulation of species (particles, molecules, etc.) stopped on the surface of the membrane. Considerable debate exists regarding the suitable theoretical work to describe this phenomenon. The theoretical work describing the phenomenon has been the subject of much controversy. The effects of various parameters on permeate flux decline and the mechanisms of membrane fouling have been investigated. However, little progress has been made in understanding the fundamental mechanisms of membrane fouling. Therefore, efforts have been made towards the theoretical modelling of the fouling phenomenon as well as towards the prediction of flux and rejection as a function of time. The complexity of these models is reduced by using numerical simulations. These CFD simulations yield a better understanding of these complex processes and help minimize the number of experiments. As a consequence, CFD has become an effective tool to achieve the goal of a better design more rapidly and cost effectively. The impressive improvements in computer performance have been matched by developments in numerical and mathematical methods [2]. As computation opens new application areas, it stimulates new ideas for mathematical and physical modelling and algorithms. Experimental studies provided only qualitative or perhaps semi-quantitative information on concentration polarization, cake formation and fouling phenomena. Several techniques in the fields of nanotechnology, microfluidics, optics, spectroscopy and sensors need to be refined in order to improve their accuracy and resolution. The relative complexity of these flow problems highlights the necessity to understand the physics and mathematics of the underlying problems. CFD offers the possibility to model many situations, provided suitable computing power is available. Fluid transport phenomena are of great importance for membrane separation systems. They occur while the various separation mechanisms are at work. A membrane module is the housing for a membrane that physically seals and isolates the feed stream from the permeate flux. The performance of a module is inherently linked to the fluid movement through its volume, and thus its geometrical configuration is an important design consideration. The system will take into account the flow, the bulk region of

the module, the boundary layer transport, the diffusive transport and the separation activity driven by the surface wet-ability, as well as the electrostatic, chemical and physical interactions at the membrane surface. All these processes are interdependent and non-linear. A CFD approach is suitable for combining them into a numerical simulation. Commercial CFD packages are often used to make such numerical studies. The attraction of these methods is that they make it possible to see the effects of different parameters on the system performance. Models exist which allow the analysis of the effects of the variable permeation, rejection changes, variable physical properties of the solution being processed and gravitational influence. Many of the models, although simplified, have proved to be remarkably accurate at predicting membrane performance over some limited ranges. 2.2. CFD equations under laminar conditions The basic transport equations that govern the flow of a viscous fluid are mathematical representations of conservation principles under laminar or turbulent conditions. Specifically, they represent the conservation of three physical quantities: mass, momentum and energy. The principle of mass conservation applied to a flowing fluid result in the equation: ∂ρ + div(ρυ) = 0 ∂t

(1)

where ρ (kg m−3 ) is the density and υ is the x component of the velocity (m s−1 ). The principle of conservation of linear momentum dictates that, for any fluid element: ∂(ρυ) ∂P + div(µ grad υ) + SMy + div(ρυu) = − ∂y ∂t

(2)

where P is the pressure (Pa), µ the viscosity (Pa s) and SMy the source term in the y direction. All these equations are solved by different programs under laminar and turbulent conditions. It is theoretically feasible to use the governing equations described in this section for turbulent as well as laminar flows. The major difference between the simulations of turbulent flows and laminar flows is the manifestation of eddy motions with an extensive range of length scales in the turbulent flow regime. However, today’s computers are not fast enough and they do not possess the required storage capacity to solve these equations directly. The reason is that turbulent motion contains scales that are typically 103 times smaller than the extent of the flow domain. To resolve the motion on these scales by means of a numerical procedure requires a mesh discretization beyond the capabilities of modern computers. Even though it is unreasonably hard to apply the governing equations to turbulent flows, it is possible to describe turbulent motion in terms of time-averaged quantities and to use the resulting descriptions in the conservation laws for mass, momentum, and energy. The main consideration of the time-averaged method is that time dependent flow properties can be described as the sum of a time-averaged steady mean component and a time-varying

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fluctuating component. These equations could be written as ∂ρ + div(ρU) = 0 ∂t

(3)

where U is the average x component of velocity (m s−1 ): ∂(ρU) ¯ + div(ρU U) ∂t


 ∂P ∂(ρu 2 ) ∂(ρu υ ) =− + div(µ grad U) + − − + SMx ∂y ∂x ∂y (4)

where U¯ is the average velocity vector (m s−1 ), u the timeaveraged fluctuating x direction velocity (m s−1 ), v the timeaveraged fluctuating y direction velocity (m s−1 ) and SMx is the source term in the x direction. All of these equations represent the laminar and the turbulent conditions. There are several turbulence models that could be used to solve these equations. Conventional models include the mixing length, k–ε, RNG k–ε, Reynolds stress and algebraic stress models. Recent models include the LED (large eddy) method which is based on spacefiltered equations. 2.3. CFD for laminar conditions Many authors are interested in the simulation of laminar conditions since most of the filtration processes work under these conditions [3–37]. The first simulations of flow in a membrane were undertaken under laminar conditions in channels with porous walls [3,4]. Laminar conditions were studied first as it is easier to generate flow in the membrane. Later, Friedman et al. [5] considered the effect of the viscosity in their computation. The enhancement of these models culminated with the investigation of laminar flow in a porous pipe with variable wall suction or variable radial mass flux or numerical solution of micropolar fluid in porous wall [6,7]. A summary of the recent developments on the role of fluid mechanics in membrane filtration was made by Belfort et al. [8,9]. They described several attempts to model both steady one and two-phase flows as well as unsteady flows. Many authors are very interested in using this method to optimize membrane processes [11–16]. For example, Karode [11] studied the pressure drop for a fluid flow in a rectangular slit and cylindrical tube with porous walls. They assumed the wall permeability to be constant. The pressure drop prediction using their expression for constant wall permeability was within 2–10% of the numerical CFD solution. The accuracy of the numerical routine was verified by comparing numerical predictions for constant wall velocity using Berman’s solution. The agreement was best for low recoveries. For example, in the case of ultrafiltration–microfiltration, where the channel pressure drop would be a significant fraction of the inlet pressure, the expressions presented in their work for constant wall permeability could be more accurate. Marriot et al. [12,13] worked on a more general approach to model hollow-fibre and spiral-wound membrane modules. The model was developed from rigorous mass, momentum and energy balances. The high precision of

Fig. 1. Velocity field in a shell and tube system with inner tube wall permeability in the radial direction [14].

the results presented revealed the importance of this complete study. This new CFD approach was verified by application to gas separation, pervaporation and reverse osmosis case studies. Their work is to be distinguished from most existing models, which are usually process specific and are only applicable within a restricted operating range. Simulation results for these systems agreed with experimental data. Their very interesting paper demonstrated the generality of the detailed model by considering a range of different membrane systems. Moreover, since these early works, better performing solvers have been developed to improve computational fluid dynamics. They use more equations, as they use both the Navier–Stokes and Darcy’s equations. The Darcy equation is J = Lp (TMP − σπ)

(5)

where TMP is the transmembrane pressure (Pa), J the permeate flux (m s−l ), π the osmotic pressure difference across the membrane (Pa), σ the Stavermann reflection coefficient, Lp the permeability (m s−1 Pa−1 ). Nassehi et al. [14] used Darcy’s equation to represent the porous wall conditions. The flow tangential to the porous tube surface was modelled by the Navier–Stokes equations. They developed a new method for the imposition of permeable wall conditions in viscous flow domains. It was based on the finite element method and was more robust because they added more complex governing equations and boundary conditions. This method offers a means of linking of the free flows modelled by the Navier–Stokes equations to the flows through permeable walls described by the Darcy equation. This technique was applied to model the feed stream and permeate flux in an axisymmetric case. The results allowed visualization of a complex flow in a porous tube. Since this work, we can for example visualize and evaluate the pressure loss in a porous tube more easily (Fig. 1). Thus, this model for crossflow filtration can be considered as the first step towards creating a complete model. The attraction of these semi-analytical models is that it is possible to tailor them for relatively quick and easy investigation on the effect of different parameters on the performance of a membrane system. Accurate modelling of the flow and concentration polarisation in pressure-driven membrane processes is inhibited by the complex couplings between the flow equations and the variable solution properties. Das et al. went on developing this model a few years later [15]. Their results showed that the model developed can predict the patterns expected for flow and mass transport systems involving a porous wall. The velocity field and

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Fig. 3. Geometry of a crossflow unit [18].

Fig. 2. Representation of laminar flow in a tubular membrane with a cylindrical coordinate system [16].

the flow circulation near the interface for different permeability were described. Their model provides a robust mean of analysis of the flow hydrodynamics. Another very interesting study was performed by Damak et al. [16]. They worked on a laminar, incompressible and isothermal flow in a cylindrical tube with a permeable wall. They assumed the flow to be axisymmetric. As Das et al. did, they used the Navier–Stokes equations and Darcy’s law to explain the transfer in the tube and in the porous wall. Moreover, with an adequately long entrance region without upstream suction, the tube was considered to generate a flow that is fully developed at the porous tube entrance. Furthermore, the classical assumption of constant physical properties of the system, such as density, viscosity and porosity of the wall was made in order to simplify the problem (Fig. 2). The flow in the free fluid region and in the porous medium in the radial direction was described by the Navier–Stokes equations and Darcy’s law, respectively. First, the model was carefully verified. Then, various simulations based on different axial Reynolds numbers and filtration numbers were presented. The results were discussed to study the validity of classical assumptions made in simplified models. Moreover, the relative divergence between the axial velocity profile with and without wall suction was characterized. Therefore, the development of mathematical models has offered a theoretical structure for the understanding of the phenomenon responsible for flux decline. All the recently reported works regarding the simulation of filtration processes are based on these models. This gives the opportunity of simplifying all the existing theories and of understanding these phenomena better. We consider that these works are a good beginning for the modelling of fluid flows in membrane processes under laminar conditions. Many specific cases could be treated in laminar conditions. Progress could be made towards the development of more accurate models. 2.3.1. Cake and polarisation–concentration models Several methods have been used for in situ monitoring of concentration polarization in order to better appreciate the physicochemical processes governing the growth of a polarized layer of solutes near a membrane surface. They include light deflection techniques, magnetic resonance imaging, direct pressure measurements, direct observation through the membrane, direct visualization above the membrane, laser triangulometry and optical laser sensors [17]. But, despite the potential of these techniques to advance the understanding of membrane filtra-

tion, the investigations available are quite limited in scope. The studies published provided only qualitative or perhaps semiquantitative information on concentration polarization, cake formation and fouling phenomena. Several of these methods need refinements to improve accuracy and resolution. This is the reason why computational fluid dynamics is so often used to develop and understand these processes. Simulations are more precise and easier to perform than experimental studies but require the input of suitable constitutive relations for quantities such as viscosity, diffusivity, rejection coefficient, etc., which are not always known to estimate the cake thickness, the rejection or the profile of the flow. For example, Geissler et al. [18] developed a dynamic model of crossflow microfiltration in flat channel systems under laminar flow conditions, often used in bio-technological processes. The model was based on the description of the hydrodynamics in channels with rectangular cross section, permeable walls and some empirical parameters. This kind of simulation is suitable for estimating the necessary membrane area for the filtration process (Fig. 3). Their numerical code is based on a finite volume method, which solves the Navier–Stokes equations. The authors’ theoretical predictions were verified by measuring the local permeate flux and local cake height during the non-stationary phase of crossflow filtration. They verified their computation with experimental results. The module was made of Plexiglas, which allowed visualisation of the mechanism of particle deposition and measurement of the filter cake height with a camera. The simulations were made in several cases and the authors concluded that for absolute stationary values the computation agreed with the experimental results within a range of 15% for all parameters and suspension systems studied. The model developed in their paper is nearly suitable to predict the course of a filtration process in a flat channel crossflow unit and to estimate the cake thickness quite well. A more precise model was then developed by Lee and Clark [19] who worked with a standard type of channel used for crossflow ultrafiltration. From the study of dead-end filtration of monodisperse colloidal suspensions, they succeeded in showing that when the particle size decreased, the specific cake resistance increased. However, it did not increase as predicted by the Kozeny–Carman equation. They also showed that an increasing transmembrane pressure resulted in a denser cake layer, which increased the specific cake resistance. Their goal was to write a law that could be applied to describe the effect of transmembrane pressure on the specific cake resistance more accurately. The numerical model of cross flow filtration developed in their paper successfully explained the fundamental mechanisms involved in flux decline during crossflow ultrafiltration of colloidal suspensions. The model provided a useful tool for investigating the

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effect of various operating parameters, such as the particle size, feed concentration, axial velocity, and membrane dimensions. That study was made possible by the numerical simulations, which allowed a reduced number of experiments. The model required the specific cake resistance as an input parameter, which was obtained independently from the dead end filtration tests. The simulations showed that the model predictions were in good agreement with crossflow experimental results. Carroll [20] developed a model to demonstrate these mechanisms for typical hollow-fibre membranes during the dead-end microfiltration of particulate suspensions. The model accommodated both compressible and incompressible cakes, and predicted highly localised fluxes, which are completely absent in planar membranes. These localised fluxes combined with transmembrane pressure gradients to produce non-uniform cake growth patterns and non-uniform cake resistance profiles. The role of cake properties and operating conditions in determining the permeate flux decline mechanism was established from the evolution of the flux profiles. The author deduced from simulations a flux decline model incorporating cake compressibility for microfiltration through a hollow-fibre membrane. This new localised flux decline mechanism in hollow fibres has a number of important consequences. Firstly, it is important for establishing the role played by hollow fibre properties in the fouling of hollow fibre membranes. The rate of flux decline with throughput depends upon the properties of the fibres and of the cake. Secondly, the model facilitates predictions of flux decline for hollow fibres from the flux decline for planar membranes. Finally, it is essential to consider this localised performance in the situation of more complex crossflow systems. The non-uniform patterns of particle deposition in hollow fibres and local variation in convective forces could also influence the rate of cake accumulation in such systems. All these studies show the important role that CFD can play in understanding cake formation. The models created provide a useful instrument for studying the effect of different parameters, such as the particle size, feed concentration, axial velocity and membrane dimensions in many types of filtration. However, all of the studies developed in these papers focused on particular cases. We consider that the fouling mechanisms are too complicated to be simulated successfully for each case (different type of membrane, solution, filtration, operating conditions, etc.). Many parameters have to be taken into account. The model has to solve the Navier–Stokes equations for the suspension flows as well as the convection diffusion equations, taking into account the effect of the shear rate. Moreover, existing models consider the pressure loss along the membrane using the Hagen Poiseuille law, which is an important approximation. Therefore, the determination of a general model representing these phenomena seems difficult to achieve. 2.3.2. CFD and models of flow through the porous material There have been many studies on flow through a porous membrane [21–28]. Experimental studies cannot be complete because sensors cannot be installed inside the porous material. Therefore, computation is an attractive approach to better understand the behaviour of the liquid through the membrane. Moreover, analytical solutions to the convection–diffusion mass

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Fig. 4. Mesh generation in the normalized fluid flow channel [21].

transfer equation are very difficult to obtain. Several attempts have been made to solve this problem numerically by finite difference methods to predict the concentration profile within the flow channel. Under an applied pressure, water penetrates through the membrane pores and the solute is rejected by the membrane. As a result, a thin concentration boundary layer is created near the membrane surface and the flux declines. Huang and Morrissey [21] tried to better understand this phenomenon of polarization. The purpose of their work was to use a numerical model to reproduce the process of concentration polarization and to model the solute concentrations on the membrane surface. They aimed to create a method that could be used in the engineering analysis and design of crossflow ultrafiltration processes. They used a finite element method and meshed the surface of the membrane with a very fine mesh to capture the important physical processes (Fig. 4). Specifically, when a very small diffusion coefficient is examined, substantially finer meshes must be used in the vicinity of the membrane surface to cope with the steep concentration gradient. Using CFD, the authors illustrated the way in which the diffusion coefficient influences the thickness of the concentration boundary layer. They showed that for a given permeate flux, there was a linear relationship between diffusion coefficient and thickness of the concentration boundary layer, which is in agreement with classical theory. This model can be used to predict the mass transfer coefficient, which is an important parameter in the creation and analysis of a membrane filtration process. Their work also highlights the need for extremely high resolution near the membrane surface since the boundary layer is very thin. Richardson and Nassehi [22] developed an algorithm based on the Streamline Upwind Petrov Galerkin finite element scheme for the solution of flow and convective dispersion equations in two-dimensional domains with solid and porous walls. Their approach was shown to provide a method for reproducing concentration profiles within geometrically complex domains

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Fig. 5. Solution domain with a flat porous wall opposite a curved porous wall [22].

(Fig. 5). The characteristic of this approach was the construction of a simple technique for the treatment of concentration boundary conditions along the porous walls. Therefore this model offered a useful step towards the development of a fast engineering technique for the quantitative analysis of crossflow filters. The applicability of the model was established in a large range of domains comprising both flat and curved porous walls. Moreover, an important step was made because they showed that the model was able to take into account the link between the physical and rheological factors and the fouling on the permeable segment, and thus the process performance [23–25]. Chatterjee et al. [26] developed a numerical solution, based on the finite difference method, for modelling the performance of a radial flow hollow fibre reverse osmosis module. The three-parameter Spiegler Kedem model was used to describe the mass transport through the membrane. The membrane performance factors as described by the three-parameter Spiegler Kedem model were approximated from experimental data. Fig. 6 shows how both the retentate and the permeate flows pass through a module and how the finite difference grid is used. By comparing these results for different NaCl concentrations, they considered the solution diffusion model adequate for describing the mass transport across the membrane, the parameters evaluated from the present model being not very different from the results obtained by the model Sekino et al. [27,28]. More experimental data for phenol separation were obtained on a hollow fibre module and the model was used to analyse the data. Moreover, the CFD results clearly showed that the two parameters model used by Sekino et al. may not be sufficient for accurate design and analysis of many solute–membrane systems. Furthermore, this parameter estima-

Fig. 6. Finite difference meshes construction for a hollow fibre module [26].

tion program can also be used for developing the mass transfer correlation in the radial flow hollow fibre modules. These studies show that models can be created to predict the operations for ultrafiltration, microfiltration, nanofiltration and reverse osmosis with good accuracy. Moreover, the CFD models do not neglect the physical and rheological factors nor the properties of the foulant on the permeable wall. Very fine meshes seem necessary to achieve high accuracy. This requires great computer power and long calculation times, which could be a handicap in enhancing membrane processes. 2.3.3. CFD and osmotic pressure: nanofiltration and reverse osmosis Advances in the understanding of membrane transfer processes and their optimisation require models that can simulate all the important physical processes occurring in membrane systems [29–37]. The operating osmotic pressure is a basic parameter governing the design and evaluation of a membrane separation system. For example, the performance of a spiral-wound module can be improved by optimizing some key geometrical parameters for given operating conditions. Many authors are interested in this application. For example, Ben-Boudinar et al. [29] made numerical simulations of a spiral wound module. The investigation covered a wide range of feed conditions by using experimental data provided from two different types of modules. In the case of the desalination of sea water, their program facilitated the evaluation of the concentrations, pressures and flow rates in the feed and permeate channels at any point in the module. For illustration, they decided to work with two typical modules. The results were obtained with maximum deviations of about 10% and 15% for the permeate flow and concentration, respectively. They suggested that a better calculation could be achieved if a more precise relationship for the mass transfer coefficient could be established. Many authors assume that spiral-wound modules typically used in nanofiltration and reverse osmosis systems can be considered as a succession of channel slits. This slit-type configuration represents the basic model to be studied and characterised for the design and optimisation of such industrial membrane modules [30–35]. Such a configuration strongly restricts the circulating flow rates to low values, typically giving rise to laminar flows. Fig. 7 shows a typical geometry used in this modelling approach. When the slit is an open channel, the laminar flow becomes fully developed in its first stages, as concluded by Geraldes et al. [32], leading to a very high value of the mass transfer resistance or to severe problems of concentration polarisation, as a consequence of the growth of the concentration boundary layer.

Fig. 7. Scheme of the NF slit used for the numerical predictions, with uniform inlet velocity and inlet solute concentration profiles [31,32].

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Overcoming these effects is important for the optimisation of mass transfer. In fact, mass transfer in slit-type channels can be improved by modifying the hydrodynamic parameters and the design and by decreasing the concentration at the boundary layer. Moreover, the feed flow influences the growth of the boundary layer and its mixing with the main stream. As a result, the design of membrane modules needs accurate analysis of the concentration polarisation phenomenon. Nanofiltration membrane performance can be predicted in terms of either apparent rejection coefficients or permeate fluxes resulting from CFD or a mass transfer model using boundary conditions that take into account solute–solvent–membrane interactions. For the CFD predictions, the variation of the solution properties of density, viscosity and diffusivity with the solute concentrations were considered. The previous mathematical model described the flow and mass transfer in the feed near the membrane, which acts as a selective barrier. The effect of concentration polarisation on the decline of the permeate fluxes was also taken into account in the predictions of these fluxes. A correlation for the ratio between the concentration and the hydrodynamic boundary layer thickness was then obtained, being valid for the ranges of interest in nanofiltration and excellent agreement was obtained with experimental values. Such a correlation gave some insight into the different mechanisms involved in the growth rates of those boundary layers. The use of CFD for the modelling of transport processes occurring at the fluid phase adjacent to a nanofiltration membrane allowed the evaluation of the concentration polarization effect. Furthermore, the calculation of intrinsic rejection coefficients, which are linked to the membrane characteristics and to the operating conditions in terms of transmembrane pressure and feed solute, concentration could be determining. Wiley and Fletcher [36] proposed a general purpose CFD model of concentration polarisation and fluid flow. The model has undergone extensive verification and highlights also the need for very fine meshes near the wall and suitable high order numerical schemes, especially when the polarisation is high. The problem is that very fine meshes necessitate powerful computer and necessitate high cost investment in term of time calculation. The need for selfconsistent physical property models was highlighted. The CFD model developed used explicit equations for property variations, and it can now be easily adapted to include any arrangement of variation in wall flux, rejection, viscosity and diffusivity. The flexibility of this model was only limited by the ability of the user to correctly describe the physical properties. The equations were solved using the finite volume code CFX4. The model was extended to examine the consequence of buoyancy in the salt water system under reverse osmosis conditions [37]. In this case the need to choose the location of the boundary condition carefully to ensure that it does not cause non-physical flow was highlighted. Application of the model to brackish water revealed that gravitational effects are only significant at very low flow rates. The authors also showed that in the reverse osmosis system the extraction rates are not sufficient to generate density differences between the feed concentration and the fluid at the surface of the membrane, which can cause significant buoyancy effects in horizontal flat channel systems. These models show a trend towards greater completeness, as not only do they include

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the variable fluid properties but also explicit the modelling of the membrane, making CFD an increasingly powerful tool. These simulations represent a powerful tool for enhancing nanofiltration and reverse osmosis processes and the models become more and more precise. 2.4. CFD under turbulent conditions Most studies have been done for laminar conditions but for some researchers it proved necessary to work under turbulent conditions [38–42]. For example, ultrafiltration modules operate in turbulent regime and the mixing properties of the turbulence are very important to minimize solute concentrating in the near wall region. Pellerin et al. studied the turbulent transport in membrane modules by CFD simulation in two dimensions [39]. The CFD package used was suitable to incorporate separation models based on the chemical and physical interactions between specific membranes and feed solutions for comparison with experimental results. The pressure-related boundary condition for permeate velocity is an essential characteristic for enlarging the model to multi-component systems where the membrane separation performance would be modeled. To this effect the convection–diffusion equation for solute transport was added and the effect of inlet concentrations and diffusion coefficients were studied. Using CFD, they demonstrated that the most important parameters were the Reynolds and Schmidt numbers. Both dimensionless numbers were found to play a key role in controlling the solute concentration gradients. An important work regarding modelling of concentration polarisation and depolarisation with high frequency backpulsing was reported by Redkar et al. [42]. In this study, a turbulent back transport of rejected solutes from the deposited layer to the bulk was ensured by the action of stirring. In fact, the gel started appearing only when the membrane surface concentration exceeded the solubility limit of the solutes in the solvent. Therefore, this deposited layer was not a true gel layer but may be called a gel-type layer whose properties are supposed to be the same as those of the cake in filtration equipment. As the membrane rejects the solutes, the thickness of this deposited gel layer increases. But at the same time, due to the stirring, some solutes are removed from the cake surface and go into the bulk. The Kozeny Carman equation was used to relate the permeate flux with other operating variables. So this work was a combination of the filtration theory and a gel polarisation model. This model depended on three main parameters, namely the back transport coefficient, permeability coefficient and membrane hydraulic resistance. They gave the predicted and experimental values of the maximum fluxes and the corresponding optimal forward filtration times. The optimal forward filtration time increased with increasing backpulse duration, whereas the maximum flux was relatively insensitive to the backpulse duration. Most importantly, the fluxes obtained with backpulsing theory predicted that the forward filtration time which maximizes the flux is, in general, slightly longer than the critical value which leads to cake or gel formation. The small number of papers focusing on turbulent conditions tends to prove that modelling in turbulent conditions is difficult. The lack of a turbulent model is significant in this case and researchers still

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do not agree on the validity of the current commercial models proposed. In fact, it requires the resolution of more complicated equations and very fine meshes. Consequently, the use of more powerful computers is necessary and the computing times should be longer. However, many authors are getting attracted to model turbulent conditions because many membrane processes work under high Reynolds numbers. Several papers have recently been published on this subject and referenced in this part, which could lead to opportunities in membrane processes development. 3. CFD and hydrodynamic conditions The use of CFD allows us to determine, describe and optimize the complex hydrodynamics generated by pulsatile flow and gas sparging [43–53], spacers [54–65], Dean and Taylor vortices [66–79] and geometry [80–84]. The performance of pressuredriven membrane processes is limited by factors like concentration polarisation, cake-layer growth and fouling, resulting in low volumetric permeation rates. In recent years extensive studies have been performed – modification of the membrane surface, optimization of scouring and hydrodynamic methods – for cancelling or reducing these limitations and thus improving membrane processes. In fact, an increase in the shear stress induces a decrease in the cake layer thickness and generally creates a greater permeate flow. Various approaches, including generating turbulent flows, placing inserts in the flow channel, creating turbulence promoters and pulsations of feed flow, have been used to improve the migration of the solute or particles from the membrane surface to the bulk flow. For example, to limit the particle layer growth, high shear rates can be caused by rotating parts of the module. Several studies have shown that fluid instabilities like Taylor vortices, established in a rotating annular filter, are very effective in reducing concentration polarisation and particle deposition due to high shear rates. These shear rates increase rapidly with the increasing axial feed flow and lead to enhanced mass transfer between the boundary layer and the bulk phase. But, this way of optimizing the flow is not the only one. We will make a state of the art of all the technology developed to enhance these processes. We will describe the gas sparging and Dean vortices to illustrate respectively the pulse-gas sparging and Dean and Taylor vortices. 3.1. Pulses and gas sparging The improvement of crossflow ultrafiltration has been studied by many authors who injected air in the concentrate inside the membrane [43–53]. The use of unsteady flow can lead to significant benefits by increasing heat and mass transfer rates. It was demonstrated that the permeate flux increased with pulsatile flow and that the accumulation of the retained particles at the membrane surface was reduced very effectively, probably mainly through the increased shear. It has already been proved that injecting air could diminish the concentration polarisation layer during ultrafiltration of macromolecules. The development of the filtration flux was supposed to be due to an increase of the turbulence and also of the crossflow velocity inside the

membrane. This theory has been demonstrated in several studies [43–51]. This way of creating turbulence by injecting gas allows creation of a gas liquid two-phase crossflow operation that can significantly increase the permeate flux and, moreover, can improve the membrane rejection characteristics. This idea was developed by Stanton et al. [52]. In fact, injecting air bubbles into the liquid feed to generate a two phase flow stream has proved to be an effective, simple and low cost technique for enhancing ultrafiltration processes and has allowed better rejection. Previous studies had been based on the analysis of the experimental data and mass transfer correlations. The authors attempted to model the slug flow ultrafiltration process using the volume of fluid method with the aim of understanding and quantifying the details of the permeate flux enhancement resulting from gas sparging. For this numerical study, the commercial CFD package FLUENT was used. This CFD was used to simulate the motion of a single Taylor bubble rising in a liquid flowing through a circular cross-section tube. In vertical pipes, Taylor bubbles are axisymmetric and have round noses, while their tail is generally assumed to be nearly flat. The Taylor bubble occupies most of the cross-sectional area of the tube. When the bubble rises through a moving liquid, the liquid that is flowing ahead of the nose of the bubble is displaced as a liquid film. It starts flowing downwards in the annular space between the tube wall and the bubble surface. Alongside the bubble, the liquid film accelerates until it reaches its terminal velocity, provided the bubble is long enough. At the rear of that bubble the liquid film plunges into the liquid plug as a circular wall jet and produces a highly agitated mixing zone in the bubble wake. It is generally believed that a kind of vortex exists in this mixing zone. This region is believed to be responsible for mass transfer enhancement due to the increase in the wall shear stress (Fig. 8). The authors thought that it is essential to understand and quantify the details of slug flow dynamics and to identify their effect on ultrafiltration performance. In their paper, they explained that permeate flux enhancement was due to gas sparging by examining the hydrodynamics of gas–liquid two-phase flow and the increase in mass transfer. They showed that the enhancement can be explained by the increase in the mass-transfer coefficient. The turbulence just behind the air bubble, caused by the annular film flowing downward, is of significant intensity, and plays a pivotal role in permeate flux enhancement in tubular membranes. Furthermore, the permeate flux enhancement due to gas sparged ultrafiltration with tubular membranes can be predicted with reasonable accuracy. Taha et al. also worked on this subject [53]. They tried to develop a model using a tubular membrane module. Their representation was based on dividing the membrane surface area into various regions depending on the hydrodynamic regime in the vicinity of the membrane, as shown in Fig. 9. By analysing the fluid flow in these regions, it is possible to calculate the mass transfer coefficients for each region. The averaged permeate flux for the membrane can be assessed. The model results were compared with experimental data and with theoretically calculated data. Simulation studies were also performed to check the effects of some process factors on permeate flux development. The authors succeeded in determining three distinct mass transfer zones in the slug flow regime called falling

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domain is for researchers. This method of enhancing membrane process performance is easy to use and this was true for each case studied, as shown by the number of papers. Furthermore, the lack of recent papers tends to prove that this method cannot be optimized any further. Many improvements have already been made to enhance microfiltration, ultrafiltration, nanofiltration, reverse osmosis processes and to lead to a better understanding of these phenomena with the help of CFD. 3.2. Spacers

Fig. 8. Simulation of slug flow [52].

film zone, wake zone and liquid slug zone. The mass transfer coefficients for each of these zones were determined. With these values, the averaged permeate flux for gas-sparged ultrafiltration was determined. There was reasonably good agreement between experimental data and theoretically predicted data. The results of simulation studies show that gas sparging can be greater at higher feed concentration and transmembrane pressure. Also, increasing liquid flow rate has contradictory effects in single phase flow as well as in gas sparged ultrafiltration. These studies show the potential of computational fluid dynamics in this domain. The large numbers of papers published shows how attractive this

The term “turbulence promoters” is nowadays accepted in the literature, although the stable flows may not necessarily exhibit the characteristics of fully developed turbulence. The presence of spacers leads to increased pressure drop, as well as to the formation of localized stagnation or dead zones, where the above phenomena may be increased. Thus, the important role of membrane spacers has been recognized in the past years, and several experimental and theoretical studies have aimed at understanding the underlying phenomena and optimizing spacer configurations [54–65]. Thus, although it is common knowledge that the presence of a spacer can greatly improve the membrane mass transfer, detailed understanding of fluid dynamics in the channel has not been developed because of experimental and numerical complexities. The spiral wound membrane module configuration is one of the most common membrane configurations in the field of membrane technology applications. In spiral wound membrane modules, spacers are used to enhance wall shear stress and to promote eddy mixing, thereby reducing wall concentration and fouling. The effect of spacer filaments on flow patterns in narrow channels can just be quantified using computational fluid dynamics. To find an optimal spacer design, it is essential to determine the flow pattern and turbulence distribution in a spacer filled channel or spiral wound module. With the development of more powerful computing techniques, it has become possible to simulate flow in spacer-filled channels. Many authors have experimentally and numerically investigated flow around a cylinder between two walls. Cao et al. [54] investigated CFD simulations of net-type turbulence promoters in a narrow channel. They ran simulations in two dimensions. FLUENT was used as the CFD package to simulate the flow in a spacer-filled channel. The general governing equations were

Fig. 9. Flow regimes for two-phase flow [53].

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applied on a curvilinear grid to enable computations in complex or irregular geometries. Several turbulence models could be used to solve the Navier–Stockes equations. In the turbulent regime, the RNG k–ε model used by the authors was first used in quantum field theory. They investigated the effects of different arrangements of turbulence promoters. Spacers come in various forms that are composed of a net-like arrangement of filaments aligned parallel, transverse or at an angle to the module axis. The spacer causes flow disturbance before and after the transverse filaments. In order to elucidate the likely extent of the flow disturbance, the CFD simulations were directed towards the effect of cylindrical objects positioned normal to the flow. The simulations revealed the detailed flow patterns, velocity distributions, and turbulent kinetic energy distributions in the spacer-filled channel. It was found that the location of the high shear stress region and of the eddies was closely related to the spacer cylinder geometry and their position in the channel. By comparing simulation results, it can be concluded that careful selection of the location of the transverse filaments and of the distances between them is very important in spacer optimisation. It was found that the mass transfer differences between the top and bottom membrane surfaces could be equalised by placing the transverse spacer cylinders alternately on the top and bottom wall or suspending them in the middle of the channel. The simulation suggested that suspending transverse spacer cylinders in the channel is more desirable. The simulation also suggested that reducing the distance between the transverse spacer cylinders could reduce the distance between the shear stress peaks and produce more efficient eddy activity, which may improve mass transfer at the membrane surfaces. However, it was established that the reduction in the distance between transverse cylinders could also significantly increase the channel pressure drop and consequently increase the operating costs. The selection of optimum geometry involves a trade-off between these competing effects. Koutsou et al. [56] carried out the numerical simulation of the flow in a plane-channel containing a periodic array of cylindrical turbulence promoters. The flow in a model two-dimensional geometry, consisting of a plane-channel in which an array of cylinders is located, was studied in order to obtain a better understanding of the behaviour, the dominant features and structures, as well as the statistical characteristics of the spacers. This work was intended to get a better understanding of transport phenomena in membrane elements, where feed flow spacers tend to enhance mass transport characteristics, possibly decreasing fouling and concentration polarization phenomena while increasing pressure drop. Model flow geometry was considered, consisting of a plane-channel, in which a regular array of cylinders was inserted, acting as turbulence promoters, as shown in Fig. 10. Direct numerical simulations using the Navier–Stokes equations were performed over a range

of Reynolds numbers typical of such membrane modules. A CFD code, based on the finite-volume method, was used. During each simulation the governing equations were integrated in time by imposing a constant mean pressure gradient until the flow reached a statistically steady state. Above a critical Reynolds number of 60 the flow became unstable due to mechanisms similar to those operating in the case of a cylinder in unbounded flow. Above a Reynolds number of 78 walls eddies appeared. This characteristic of the flow in the channel wall was due to the interaction of the vortices shed by the cylinders with the vortices layers created on the channel walls. Three-dimensional effects were not considered in that study. This approach, which is common practice in turbulent flow numerical simulations, may also be useful to solve the mass transfer problem in order to obtain local mass transfer. The understanding achieved in that study is expected to facilitate optimization studies of membrane spacer configurations. Schwinge et al. [62] studied an unsteady flow in narrow spacer-filled channels for spiral-wound membrane modules. They inspected the subcritical and supercritical flow patterns in narrow two-dimensional channels for single and multiple filaments. As the cylindrical spacer filaments were oriented transverse to the main flow direction, two-dimensional CFD simulations could be used. These two-dimensional calculations are an excellent viewing device for future estimation of more complex spacer geometries. The fluid used was water at a temperature of 293 K. This fluid was supposed to be incompressible, isothermal and to have constant fluid properties. The fluid movement was governed by the Navier–Stokes equations. These equations were suitable for all Reynolds numbers, but under laminar conditions they could be solved without considering the resolution of turbulent eddies. These equations were solved using a CFD code. Five filament configurations were examined (Fig. 11). Computational fluid dynamics calculations showed complex interaction between all the parameters introduced in the numerical simulation. For a single filament in a narrow channel the wall reduced the transition, which occurred at slightly higher Reynolds numbers than for a cylinder in a free flow channel. However, the authors showed that a good adequation between process parameters and fine meshes is required. This increases the computational cost but enhances the accuracy of the results. Therefore, they suggested that calculations to three dimensions would necessitate careful and methodical identification of appropriate modelling procedures. Wiley et al. [63] explained the interest of a numerical simulation model of pressure driven membrane processes describing the selective trapping of certain particles in the feed channel and the pas-

Fig. 10. Schematic of flow geometry [54].

Fig. 11. Different configurations developed [62].

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Fig. 12. The basic shapes of commercial net spacers [64].

sage of others to the permeate channel. The effects of changes in rejection, wall permeation rates and solution properties on flow rate and concentration profiles are presented for empty channels and channels with turbulence promoters. Therefore, the authors decided to use CFD because this method can integrate the modelling of a flow in complex geometries which includes the recirculation and the difference in fluid properties between the bulk solution and the membrane wall. The finite volume code CFX4 was used to solve the continuation equations. The flow was also assumed to be steady, incompressible and laminar but the viscosity and diffusivity were likely to change. The model was confirmed with a high number of semi-analytical solutions. While the model developed has been corroborated for a series of situations, it can readily be adapted for other conditions of physical properties and of permeation rates. The advantage of the model is that it may help to estimate the effect of various assumptions on concentration polarisation. This kind of information is essential to use membrane systems properly. Moreover, their paper revealed that it was necessary to improve the CFD package used for the representation of membrane systems by including a model allowing the description of selective component exclusion. The modular nature of this validated CFD model makes it a powerful tool for the design of membrane systems. Li et al. optimized commercial net spacers in spiral wound membrane modules [64]. They decided to compare the two main types of commercial net spacers frequently used—woven and non-woven spacers, as shown in Fig. 12. A non-woven spacer can be characterized by four parameters: the distance between spacer filaments l1 and l2, the angle β between the spacer filaments and the flow attack angle α (Fig. 13). This study focused on cylindrical filaments. The diameter of the filaments (d) was half the channel height (h). The influence of the four geometric parameters on mass transfer enhancement and mechanical energy dissipation were investigated. Results from CFD simulations indicated that if l/h is small, the flow between

Fig. 13. Geometric characterisation of a nonwoven spacer [64].

Fig. 14. CFD fluid flow results [65].

two neighbouring filaments consists mainly of one “dead eddy”, which does not contribute much to mass transfer. If the value of l/h is very high l/h = 10, shedding of vortices will only enhance mass transfer in the direct neighbourhood of the filaments, resulting in lower average mass transfer. This means that an optimum value of l/h exists. The geometric parameters α and β also play an important role in enhancing mass transfer. It was found that the spacer geometry is optimal for l/h = 4, α = 30 and β = 120 over a large range of power number values. The results of the CFD simulations were validated using a few experimental data. The experiments confirmed that the geometric parameters of spacers have a considerable influence on mass transfer for a given crossflow power consumption. Comparison of the experiments carried out with different non-woven spacers showed that there is an optimal spacer geometry, which agrees with the optimal geometry obtained by CFD simulations. On the other hand, Karode et al. [65] reported results from a computational fluid dynamics study performed to visualize the steady state fluid flow structure through spacer filled channels in flat sheet form. A commercially available CFD routine called PHOENICS was used to implement the fluid flow equations. The simulation was done with a rectangular test cell showing spacer filaments and a typical grid (Fig. 14). Water was taken as the bulk fluid for all CFD simulations. These simulations were run for inlet velocities ranging from 0.25 to 1.0 m s−1 , which encompassed typical cross flow velocities in commercial membrane modules. Computational fluid dynamics simulations were run on several commercially available spacers to evaluate the effectiveness of spacers in terms of pressure drop and average shear stress. The main factors influencing the design of an effective spacer (high wall shear and low pressure drop) seem to be the ratio of filament diameter to inter-filament distance, the filament diameter and the angle between the spacer filaments. The filament diameter is an important parameter since it limits the packing density in the final membrane module. The ratio between filament diameter and inter-filament spacing influences the bulk flow pattern. The order of spacers when arranged in increasing average shear rate was found to be different for high (1 m s−1 ) and low (0.25 m s−1 ) inlet velocities. The authors compared each spacer for each case and gave advice on the choice of the spacer. Therefore, there

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where Re is the Reynolds number (dimensionless), di the internal diameter and dc the curve diameter (m): Re =

Fig. 15. Representation of Dean vortices in coiled pipe [74].

seems to be important characteristics to take into account before choosing a spacer. Spacer development is a typical application using CFD because flow or pressure sensors are impossible to insert in such a process. However, it seems that the latest studies performed in this way give the finest spacer optimization and that further progress would be difficult to achieve. In many cases, energy consumption considerations should also be taken into account in deciding on the best spacer geometry. 3.3. Dean vortices It is possible to make these vortices visible in a helically coiled membrane (Fig. 15). Experimental results show an enhanced filtrate flux and improved filtration process. The effect of Dean vortices on the filtration flux and efficiency of membrane processes in sinusoidal curved and helically coiled tubular membranes has been investigated by several research groups [66–79]. The Dean number is the ratio of the centrifugal force to the viscous force. The Dean number in a curved pipe is defined as this dimensionless parameter:  di De = Re (6) dc

ρυdi µ

(7)

Experimental studies have shown that the efficiency of crossflow filtration in curved membranes can be much higher than in straight membranes. CFD simulations were run, illuminating two important effects of secondary flow: an increase in the wall shear stress causing a reduced particle deposition and an enhanced mass transfer at the boundary layer. In addition, a detailed study of the performance of helical screw thread inserts in tubular membranes was carried out by Bellhouse et al. [66]. This study was designed to examine the complete fluid dynamic processes contributing to flux enhancement when screw thread inserts are used with tubular membranes. The computational fluid dynamics code FLUENT was used to predict the behaviour of membranes fitted with inserts. The design combined a mainly helical flow, in which Dean vortices are engendered, with an approximately axial flow, which modifies the Dean vortices into a continuous corkscrew vortex. This support created an excellent adjustment of the feed fluid and minimized concentration polarisation effects. The inserts (Fig. 16) were tested under microfiltration conditions using yeast solutions, under ultrafiltration conditions using reconstituted powdered milk solutions, and under nanofiltration conditions using synthetic dyes. The authors succeeded in showing significant boost (by factors of 6–10) of the filtration fluxes in the presence of inserts compared to the classical tubular membranes under the same operational conditions. They compared their experimental results with CFD results and good conformity was found between the CFD calculations and experimental measurements. Moulin et al. wrote a paper in which a numerical simulation of shear stress was proposed to determine the influence of the geometric parameters in four different tubes—straight, torus, helical and woven [68]. The mathematical model consisted in the resolution of the Navier–Stokes and continuity equations for a 3D steady, laminar flow of an incompressible Newtonian fluid with constant physical properties. This method consisted in creating vortices in modules without moving parts. They ran numerical simulations with nonporous cylindrical straight tubes and compared the wall shear stress at the surface of the material. They worked under laminar conditions and established several

Fig. 16. Single and three start helical screw-thread inserts [66].

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Fig. 17. Representation of the secondary velocity vectors made [74] (helical tube Re = 1, De = 0.4, θ = 520◦ , di = 3:2 mm, dc = 20 mm, b = 30 mm, water, ρ = 1000 kg m−3 , µ = 10−3 Pa s).

relationships. They concluded that Dean vortices increase the shear stress to the same extent as they enhance permeate flux in pervaporation process. They also showed that the simulation allows analysis of the flow behaviour from the laminar to the turbulent regime. Moreover, any geometry design may be simulated so that the influence of Dean vortices in straight, helical tubes as well as woven fibres can be predicted with the use of CFD. The simulations may be used to visualize the Dean vortices: their location in the flow, their centre and their rotation can be determined as shown in Fig. 17. The results obtained were in good conformity with published results obtained using imaging methods. Moll et al. developed this first concept [72]. They supposed the effect of the feed concentration was not completely understood. Their goal was to give new insight by comparing laser visualisation results with CFD results. They also used a mathematical model that consisted in solving the Navier–Stokes and continuity equations for a 3D steady, laminar flow of an incompressible homogeneous and Newtonian fluid. The algorithm which was based on the finite element method used a projection method in order to approximate the solution of the equations. In the case studied, attention was restricted to rigid, nonpermeable bounded walls. The authors compared their results with several experimental techniques of visualisation that had previously been used to reveal secondary flows. For example, Winzeler et al. [75] had used a fine aluminium suspension in a methylene blue solution and Bolinder et al. [76] had used laser Doppler velocimetry and laser visualisation in order to visualize the vortices. The numerical simulation was easy to apply and more precise than all the methods developed before. Therefore, it allowed comparison of the experimental results with the CFD simulations. Moll et al. worked at several Reynolds numbers and examined two different cylindrical tubes, straight and helical, for numerical tracers. Simulation results were considered for expanded velocity distributions and for different Reynolds numbers. They chose different sections in the channel and they obtained the positions of the particles in those consecutive sections. A trajectory was deduced from all the successive positions of a given particle. This trajectory can be anticipated in local coordinate systems, which leads to figures like Fig. 18. Considering all the results, this paper gives new insight into the effects involved in the flux enhancement

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Fig. 18. Projected trajectories of numerical tracers in a half section made by Moll et al. (helical tube, Re = 1627, di = 3.2 mm, dc = 20 mm, b = 30 mm, water, ρ = 1000 kg m−3 , µ = 10−3 Pa s, particle diameter = 0.5 m) [72].

caused by Dean vortices. As the Reynolds number increases, due to the centrifugal force, the maximum of the axial flow is transferred to the opposite of the membrane. A wide range of Reynolds numbers was treated and we can see that the axial flow was closer to the wall, which is in accord with laser visualisation. In some cases, the enhancement of mass transfer can be totally explained by the wall shear stress augmentation due to the Dean vortices. Considering this paper, when particles such as those of the baker’s yeast were present in the solution, a second effect of Dean vortices was noticeable which can be understood by considering swirling motion effects. In fact, at low concentrations, the regions where accumulation of retained matter occurs remain very thin whatever the geometry. Thus, this distribution of the accumulation is similar in straight and helical tubes. In these cases, the main mass transfer improvement is due to the averaged wall shear stress improvement. When the concentration increases significantly, gradients of retained matter are modified and the influence of the channel geometry is more and more noticeable. Therefore, in this case due to the curved pathlines, the mass transfer mechanism is different from that observed in a straight membrane: this might suggest that accumulation of retained matter occurs in a thinner zone in the helical geometry, which provides a kind of recirculation effect. These studies show the importance of numerical simulation in membrane processes to better understand these complex phenomena. Even with the most powerful sensors, we could not have obtained such precise results with experiments. Another study was performed to optimize the module design and in order to inspect the mechanisms influencing membrane filtration performance, by Bulboz et al. [77]. They focused on crossflow microfiltration in helically coiled membranes, and specifically in sinusoidal curved tubular membranes. Experimental and theoretical studies were carried out. Numerical calculations were performed to get a better perception of why Dean vortices are so efficient in limiting cake layer growth. At the beginning of the theoretical analysis, the three-dimensional velocity and pressure field were calculated by solving the Navier–Stokes equations numerically in a three-dimensional Cartesian coordinate system with a finite control volume method for laminar fluid flow in a helically coiled and a sinusoidal curved tube with impermeable walls. The solu-

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Fig. 19. Geometry of the sinusoidal curved tube and streamlines of the velocity field in the cross-section after three 180◦ curvatures developed [77]: (a) De = 100, Re = 500; (b) De = 400, Re = 2000; (c) De = 600, Re = 3000.

tion of the governing equations was achieved for the case of an incompressible, isothermal, fully developed and steady state Newtonian fluid flow. The geometry of the tube is described in Fig. 19. This simulation allows using different crossflow velocities for different solute concentrations, Dean numbers and Reynolds numbers. The beneficial effect of a curved membrane can be explained by an increase of the particle lift force, which can be correlated with the wall shear stress. The rapid mixing between boundary layer and bulk phase, results in an improved mass transfer in the cross section. Further progress in modelling the filtration process should be made to reach a better understanding of the flow and mass transfer in curved membranes. Based on these results and on future results, the design of new optimized crossflow microfiltration modules could be proposed. The authors also can suggest optimum parameters for the filtration of a latex and yeast suspension solution. The results of these simulations corresponded to the results of the experimental investigation, which showed that the highest flux improvement occurred for Reynolds numbers about 2000 and Dean number about 400. They also demonstrated that curved membranes are more effective than straight membranes. They were able to design a new type of optimized membrane more effective than a straight membrane. All these developments in modelling the filtration process allow a better understanding of flow and mass transfer in curved membranes. The efficiency of this approach for enhancing membrane processes is evident for large numbers of papers published. Unfortunately, manufacturers do not use this approach to improve the performance of these filtration processes.

ing barrier. The performance of a membrane separation process can be characterised by the product of the permeate flux and the total filtration area. These kind of developments interest many researchers [80–84]. A very interesting paper was written by Tarabara et al. [81]. This paper showed that the geometry has an important place in the membrane enhancement. They presented results of computational fluid dynamics modelling of the flow within a crossflow membrane filtration cell whose dimensions replicated those of the SEPA CF. They used a commercial CFD package FLUENT based on the finite-volume method. Flow field was characterized in terms of velocity, pressure, and shear stress. Flow was found to be unidirectional over the greatest part of the channel area with the exception of the corners of the channel. Stagnation areas were observed in dead ends at the inlet and outlet of the channels. The relation between the highest shear rate created in this geometry and the average inlet velocity was given. This paper showed that it was highly possible to enhance membrane geometry by using CFD. Dolecek et al. carried out many studies in this domain [82,83]. Their first study dealt with the modelling of permeates flow in a multichannel ceramic membrane. The aim was to simulate numerically the permeate flow in porous body of a multichannel membrane element and to estimate the effect of the element configuration on the flux performance. It was limited to pure permeate flow simulation, leaving the simultaneous filtration for future work. Permeate flow in the porous body of a multichannel membrane element was replicated numerically using the finite element method. A simple honeycomb configuration was chosen to reveal the effect of geometry parameters on the performance. The geometry of a ceramic membrane element was chosen for the simulation of the permeate flow. This simple rectangular honeycomb configuration is used, for example, in ceramic membrane monoliths made by CeraMem Corp (Fig. 20). The authors wanted to check the influence of many parameters. The parameters that influence the performance of a membrane element are: permeability, permeate viscosity, flow channel pressure, permeate conduit pressure,

3.4. Geometry One of the most important characteristics of these membrane processes is the compactness of the membrane. The exchange surface is large, which enhances filtration. For example, inorganic membranes are manufactured especially in flat-disc, tubular or multichannel configurations. The channel surface is covered with a skin layer which has small pores and acts as a separat-

Fig. 20. A repeating section of the element [81]: (1) permeate conduits; (2) flow channels; (3) repeating section.

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distance between permeate conduits, number of flow channels between two adjacent layers of permeate conduits, porous wall thickness and flow channel width. When the permeability of the membrane element porous body is very high compared to permeability of the active membrane layer the permeate flux is controlled by the latter and the flux performance is proportional to the membrane packing density. When the permeability of the active membrane increases the effect of the membrane element configuration becomes significant. Generally, for a given value of the permeability the maximum permeate flux per unit volume of a membrane element can be found. When the distance between permeate conduits decreases the active membrane surface increases, which affects flux performance positively. On the other hand, for very short distances, the cross section available to the permeate flow decreases, affecting the permeate flux negatively. When the number of channels increases, the membrane surface increases but at the same time the permeate from inner channels must travel a longer distance to reach a permeate conduit. Thus the contribution to the overall flux is relatively small from the inner channels for high numbers of channels. There are dead zones of small pressure gradient around the inner channels. This work can help to develop new membrane geometry. A more complex study was performed by the same authors two years later. Their goal was to develop configurations with the highest possible filtration area per unit of membrane volume. The filtration area packing density of inorganic multichannel membranes can reach 400 m2 /m3 . The purpose of this study was to extend the results to the simulation of permeate flow in the porous support of inorganic 19 channel membranes—probably the most used multichannel configuration. This configuration is also more suitable for an experimental verification of numerical results. The effects on the permeate flow distribution of the membrane geometry and of the ratio of skin layer to support permeability were studied. The results of numerical simulations can be used both for filtration and backflush operating modes. It was found that channel contributions depend significantly on the ratio of the skin layer to porous support permeabilities and to a lesser extent, on the membrane geometry. The authors proved that the enhancement of the multichannel membrane performance is not generally proportional to the increase in filtration area packing density. If the permeability of the porous support is much greater than that of the skin layer the permeate flow is controlled by the latter and the contributions of all the channels are uniformly distributed. This study showed that channel contributions to total permeate flux in inorganic 19 hexagonal channel membranes depend significantly on the ratio of skin layer to porous support permeabilities and to a less extent, on the membrane geometry (Fig. 21). For low values of the permeability, the permeate flow is governed by the skin layer resistance and the contributions of all the channels are nearly equivalent; in this case, the membrane performance is proportional to the filtration surface. For high values of the permeability, a region of nearly constant pressure formed in the internal part of the membrane and contributions of middle and central channels were significantly smaller than those of corner and wall channels because the transmembrane pressure was lower in the middle of the membrane. Therefore manufacturers have to take this into account when they produce

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Fig. 21. Membrane geometry [82].

new membranes. There is no use in placing a lot of channels in the membrane to increase its surface if the inner channels do not work. On the other hand, if the permeability of the skin layer is high or if the channels are not covered with the skin layer at all, then the channels near the membrane surface contribute significantly to the total permeate flux while the contributions of the inner channels are negligible (Fig. 22). In this case, the permeate flow is controlled predominantly by the porous support permeability. Moreover, another parameter is to be taken into account: the authors called it coefficient α. In fact, this parameter represents the ratio between the flow channel width and the inter channel wall thickness. This review is very important and shows that it is essential to respect a minimum width between two channels. This minimum width allows a minimum of transmembrane pressure and thus the permeate in the porous can be ejected more easily. They also insisted on the fact that the larger the number of channels, the larger the surface, but at the same time the permeate from inner channels must travel a longer distance to reach a permeate conduit. Thus the contribution of the inner channels to the overall flux is relatively small for large numbers of channels. Darcovich et al. [84] proposed a new approach to develop these processes. The objective was to design a thin channel membrane module with very uniform flow properties, to use a statistical experimental design approach to determine the module parameters and to select module parameters which give uniform flow conditions over a broad range of practical operating conditions. The flow through the module channel was simulated using a finite difference code. The computational fluid dynamics scheme which calculated the above module characteristics was a turbulent transport model which used the finite difference method. Therefore, the aim of their study was to design for research purposes a membrane module with uniform flow characteristics over the permeating area. A thin channel cross flow module with a uniform flow field was designed for the characterization of flat ceramic membranes. A total of ten variables were considered for the module design. This design was used to evaluate the predicted module performance for each combination of the design variables. It was found that running the simulation at the highest cross flow velocity and the lowest module pressure always resulted in the worst

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Fig. 22. CFD results: isobars and streamlines in the porous media [82].

performance for a given module design. All these observations were made inside the porous material. With experiments, we could not have observed all of these characteristics and could not have progressed in the design of new membrane geometries. Therefore, computational fluid dynamics is an essential tool to progress in this field. Moreover, enhancing membrane processes by modifying the geometry characteristics seems an interesting approach. In these studies, the goal generally reached is to increase the wall shear stress at the membrane surface to limit the polarisation–concentration phenomenon. 4. Conclusion This review shows the interest of coupling numerical simulation and membrane processes. The large number of publications suggests the high potential of this approach for enhancing membrane processes. The increasing number of studies is clearly related to the recent developments in computer power and to the use of finer grid meshes in the vicinity of the membrane. This review shows that two approaches have been particularly considered: the comprehension of the hydrodynamics and of the mass transfer. The hydrodynamics allows the increase of the shear stress near the wall, thus allowing the enhancement of membrane processes. Commonly, the first simplifying assumptions can be summarized by: the stronger the shear stress, the weaker the fouling. Therefore, a large number of studies relate to the diphasic flows, the turbulence promoters, and to the geometries of membranes capable of generating secondary flows. The study of hydrodynamics reveals the optimum operating conditions and the most suitable geometry characteristics. However, in most cases, the membranes are created with the aim of improving the transfer of solvent. The second approach associates the models of mass transfer in the vicinity of the membrane and the hydrodynamic models. In most cases, the studies were performed under laminar conditions. Several studies were done using the models of concentration of polarization, and other studies were carried out inside the porous matrix of the membrane. Some recent studies consider turbulent flows, but the significant

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